Problem 38 - College Algebra Essentials

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$-2

Step-by-Step Solution

Answer

The solution set of the inequality \( -2 < y \leq 5 \) is the set of all real numbers between -2 (exclusively) and 5 (inclusively). This is represented graphically as a shaded region on the number line between an open circle at -2 and a closed circle at 5.

Step 1: Identify the inequality

Analyze the given inequality. Here it is provided in the form: \( -2 < y \leq 5 \). This means 'y' is greater than -2 and less than or equal to 5.

Step 2: Draw the number line

Draw a number line to represent the real numbers. Include the values of -2 and 5 to depict the given range of 'y'.

Step 3: Shade the required range on this number line

As 'y' is greater than -2 but not equal to -2, make a circle at -2 (open circle as -2 is not included) and for 'y' less or equal to 5, make a solid circle on 5 (closed circle as 5 is included). Then, shade the region between -2 and 5 on the number line to illustrate the solution set.

Key Concepts

Number Line

The number line is a crucial tool for visualizing numbers, especially when dealing with inequalities. It is a horizontal line on which every point corresponds to a real number.
Points on the left are lower in value, while those on the right are higher.
In this exercise, the number line helps to graphically display the range of values for 'y' in the inequality \(-2 < y \leq 5\).
Plotting on the number line helps to clearly show which numbers are included in or excluded from the solution set.
To start, draw a horizontal line and mark at least the boundary values, -2 and 5.
This sets your reference points for identifying which parts of the number line fall into your solution set.

Solution Set

The solution set is the collection of all values that satisfy the given inequality or system of inequalities.
In this example, the solution set includes all numbers that make the inequality \(-2 < y \leq 5\) true.
The solution set can be given in interval notation: \((-2, 5]\).
This notation indicates that y is greater than -2 but less than or equal to 5. Remember that an open interval \((-2, 5]\) or a closed interval can vary depending on the inequality signs.
Understanding solution sets involves recognizing which numbers satisfy each part of the inequality and ensuring they form a continuous group on the number line.

Inequality Graph

Graphing inequalities involves representing the solution set of the inequality on the number line.
This allows you to see the values that make the inequality true at a glance.
For the inequality \(-2 < y \leq 5\), the points show where the solutions lie.
Start by drawing a line and marking important values, then shade the section representing the possible y-values.
Ensuring that you know which numbers are within the solution set is important for understanding the scope of the inequality.The shaded part between -2 and 5 highlights where y can be, showing a clear picture of the inequality graph.

Open and Closed Circles

Open and closed circles on a number line indicate whether boundary points are included in the solution set of an inequality.
An open circle represents a number that is not part of the solution.
In this exercise, you place an open circle at -2, illustrating that -2 is not included in \(y > -2\).A closed circle shows that the boundary number is included.
For the inequality \(y \leq 5\), a closed circle on 5 signifies y can equal to 5.
Understanding when to use open or closed circles helps you accurately convey the conditions of the inequality on a graph. They work together to provide a complete visual of where y can and cannot be.